We observe that is nothing but the trigonometric interpolant
of *w*(*x*) at the equidistant points :

This shows that is in fact a dospectral projection,
which in the usual sin-cos formulation reads

Thus, trigonometric interpolation provides us with an excellent vehicle to
perform approximate discretizations with high (= spectral) accuracy, of
differential and integral operations. These can be easily carried out in
Fourier space where the exponentials serve as eigenfunction. For example,
suppose we are given the equidistant gridvalues, , of an underlying
smooth (i.e., also periodic!) function .
A second-order accurate discrete derivative is provided by center differencing

Note that the error in this case is, , no
matter how smooth *w*(*x*) is. Similarly, fourth order approximation is given
(via Richardson's extrapolation procedure) by

The pseudospectral approximation gives us an alternative procedure: construct
the trigonometric interpolant

Differentiation in Fourier space amounts to simple multiplication,
since the exponentials are eigenfunctions of differentiation,

and we approximate

Indeed, by our estimates we have for

which verifies the asserted spectral accuracy. Similar estimates are valid
for higher derivatives. To carry out the above recipe, one proceeds as
follows: starting with the vector of gridvalues, , one computes the discrete Fourier coefficients

or, in matrix formulation

then we differentiate

or in matrix formulation

and finally, we return to the ``physical'' space, calculating

or in matrix formulation

The summary of these three steps is

where represents the discrete differentiation matrix, and similarly
for higher derivatives.

__: Since (interpolation!) we
apply . How does this compare with
finite differences and finite-element type differencing?
__

__
__

__
In periodic second-order differencing we have
fourth order differencing yields
In both cases the second and fourth order differencing takes place in the
physical space. The corresponding differencing matrices have
__

__
We have seen how the pseudospectral differentiation works in the physical
space. Next, let's examine how the standard finite-difference/element
differencing methods operate in the Fourier space.
Again, the essential ingredient
is that exponentials play the role of eigenfunctions for
this type of differencing. To see this, consider for example the usual second
order centered differencing, , for which we have
The term is called the ``symbol'' of center
differencing. By superposition we obtain for arbitrary grid function
(represented here by its trigonometric interpolant)
that
It is second-order accurate differencing since its symbol satisfies
Note that for the low modes we have
error (the less significant
high modes are differenced with error but their amplitudes tend rapidly to zero). Thus we have
and this estimate should be compared with the usual
The main difference between these two estimates lies in the fact that
the last estimate
is local, i.e.,
we need the smoothness of w(x) only in the neighborhood of
,
and not in the whole interval,
.
The analogue localization in
the Fourier space will be dealt later.
__

__
Similarly, we have for fourth order differencing the symbol
In general, we encounter difference operators whose
matrix representation, __

__
In general, the symbols are trigonometric polynomials or rational functions
in the ``dual variable,'' kh, which has ``exact'' representation on the grid
in terms of translation operator (polynomials or rational functions), and
accuracy is determined by the ability to approximate the exact differentiation
symbol, ik, for , consult Figure 2.2.
__

__
__

__
__

__
Figure 2.2: The symbols of center differencing
__

Thu Jan 22 19:07:34 PST 1998